
Physica D: Nonlinear Phenomena [SJR: 1.049] [HI: 102] [3 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 01672789 Published by Elsevier [3039 journals] 
 A theory of synchrony by coupling through a diffusive chemical signal
 Authors: Jia Gou; WeiYin Chiang; PikYin Lai; Michael J. Ward; YueXian Li
Pages: 1  17
Abstract: Publication date: 15 January 2017
Source:Physica D: Nonlinear Phenomena, Volume 339
Author(s): Jia Gou, WeiYin Chiang, PikYin Lai, Michael J. Ward, YueXian Li
We formulate and analyze oscillatory dynamics associated with a model of dynamically active, but spatially segregated, compartments that are coupled through a chemical signal that diffuses in the bulk medium between the compartments. The coupling between each compartment and the bulk is due to both feedback terms to the compartmental dynamics and flux boundary conditions at the interface between the compartment and the bulk. Our coupled model consists of dynamically active compartments located at the two ends of a 1D bulk region of spatial extent 2 L . The dynamics in the two compartments is modeled by Sel’kov kinetics, and the signaling molecule between the twocompartments is assumed to undergo both diffusion, with diffusivity D , and firstorder, linear, bulk degradation. For the resulting PDE–ODE system, we construct a symmetric steadystate solution and analyze the stability of this solution to either inphase synchronous or antiphase synchronous perturbations about the midline x = L . The conditions for the onset of oscillatory dynamics, as obtained from a linearization of the steadystate solution, are studied using a winding number approach. Global branches of either inphase or antiphase periodic solutions, and their associated stability properties, are determined numerically. For the case of a linear coupling between the compartments and the bulk, with coupling strength β , a phase diagram, in the parameter space D versus β is constructed that shows the existence of a rather wide parameter regime where stable inphase synchronized oscillations can occur between the two compartments. By using a Floquetbased approach, this analysis with linear coupling is then extended to determine Hopf bifurcation thresholds for a periodic chain of evenlyspaced dynamically active units. Finally, we consider one particular case of a nonlinear coupling between two active compartments and the bulk. It is shown that stable inphase and antiphase synchronous oscillations also occur in certain parameter regimes, but as isolated solution branches that are disconnected from the steadystate solution branch.
PubDate: 20161127T16:42:30Z
DOI: 10.1016/j.physd.2016.08.004
Issue No: Vol. 339 (2016)
 Authors: Jia Gou; WeiYin Chiang; PikYin Lai; Michael J. Ward; YueXian Li
 Global dynamics for steep nonlinearities in two dimensions
 Authors: Tomáš Gedeon; Shaun Harker; Hiroshi Kokubu; Konstantin Mischaikow; Hiroe Oka
Pages: 18  38
Abstract: Publication date: 15 January 2017
Source:Physica D: Nonlinear Phenomena, Volume 339
Author(s): Tomáš Gedeon, Shaun Harker, Hiroshi Kokubu, Konstantin Mischaikow, Hiroe Oka
This paper discusses a novel approach to obtaining mathematically rigorous results on the global dynamics of ordinary differential equations. We study switching models of regulatory networks. To each switching network we associate a Morse graph, a computable object that describes a Morse decomposition of the dynamics. In this paper we show that all smooth perturbations of the switching system share the same Morse graph and we compute explicit bounds on the size of the allowable perturbation. This shows that computationally tractable switching systems can be used to characterize dynamics of smooth systems with steep nonlinearities.
PubDate: 20161127T16:42:30Z
DOI: 10.1016/j.physd.2016.08.006
Issue No: Vol. 339 (2016)
 Authors: Tomáš Gedeon; Shaun Harker; Hiroshi Kokubu; Konstantin Mischaikow; Hiroe Oka
 Analysis of the Poisson–Nernst–Planck equation in a ball for modeling
the Voltage–Current relation in neurobiological microdomains Authors: J. Cartailler; Z. Schuss; D. Holcman
Pages: 39  48
Abstract: Publication date: 15 January 2017
Source:Physica D: Nonlinear Phenomena, Volume 339
Author(s): J. Cartailler, Z. Schuss, D. Holcman
The electrodiffusion of ions is often described by the Poisson–Nernst–Planck (PNP) equations, which couple nonlinearly the charge concentration and the electric potential. This model is used, among others, to describe the motion of ions in neuronal microcompartments. It remains at this time an open question how to determine the relaxation and the steady state distribution of voltage when an initial charge of ions is injected into a domain bounded by an impermeable dielectric membrane. The purpose of this paper is to construct an asymptotic approximation to the solution of the stationary PNP equations in a d dimensional ball ( d = 1 , 2 , 3 ) in the limit of large total charge. In this geometry the PNP system reduces to the Liouville–Gelfand–Bratú (LGB) equation, with the difference that the boundary condition is Neumann, not Dirichlet, and there is a minus sign in the exponent of the exponential term. The entire boundary is impermeable to ions and the electric field satisfies the compatibility condition of Poisson’s equation. These differences replace attraction by repulsion in the LGB equation, thus completely changing the solution. We find that the voltage is maximal in the center and decreases toward the boundary. We also find that the potential drop between the center and the surface increases logarithmically in the total number of charges and not linearly, as in classical capacitance theory. This logarithmic singularity is obtained for d = 3 from an asymptotic argument and cannot be derived from the analysis of the phase portrait. These results are used to derive the relation between the outward current and the voltage in a dendritic spine, which is idealized as a dielectric sphere connected smoothly to the nerve axon by a narrow neck. This is a fundamental microdomain involved in neuronal communication. We compute the escape rate of an ion from the steady density in a ball, which models a neuronal spine head, to a small absorbing window in the sphere. We predict that the current is defined by the narrow neck that is connected to the sphere by a small absorbing window, as suggested by the narrow escape theory, while voltage is controlled by the PNP equations independently of the neck.
PubDate: 20161127T16:42:30Z
DOI: 10.1016/j.physd.2016.09.001
Issue No: Vol. 339 (2016)
 Authors: J. Cartailler; Z. Schuss; D. Holcman
 Numerical analysis of the rescaling method for parabolic problems with
blowup in finite time Authors: V.T. Nguyen
Pages: 49  65
Abstract: Publication date: 15 January 2017
Source:Physica D: Nonlinear Phenomena, Volume 339
Author(s): V.T. Nguyen
In this work, we study the numerical solution for parabolic equations whose solutions have a common property of blowing up in finite time and the equations are invariant under the following scaling transformation u ↦ u λ ( x , t ) : = λ 2 p − 1 u ( λ x , λ 2 t ) . For that purpose, we apply the rescaling method proposed by Berger and Kohn (1988) to such problems. The convergence of the method is proved under some regularity assumption. Some numerical experiments are given to derive the blowup profile verifying henceforth the theoretical results.
PubDate: 20161127T16:42:30Z
DOI: 10.1016/j.physd.2016.09.002
Issue No: Vol. 339 (2016)
 Authors: V.T. Nguyen
 A Hierarchical Bayes Ensemble Kalman Filter
 Authors: Michael Tsyrulnikov; Alexander Rakitko
Pages: 1  16
Abstract: Publication date: 1 January 2017
Source:Physica D: Nonlinear Phenomena, Volume 338
Author(s): Michael Tsyrulnikov, Alexander Rakitko
A new ensemble filter that allows for the uncertainty in the prior distribution is proposed and tested. The filter relies on the conditional Gaussian distribution of the state given the modelerror and predictabilityerror covariance matrices. The latter are treated as random matrices and updated in a hierarchical Bayes scheme along with the state. The (hyper)prior distribution of the covariance matrices is assumed to be inverse Wishart. The new Hierarchical Bayes Ensemble Filter (HBEF) assimilates ensemble members as generalized observations and allows ordinary observations to influence the covariances. The actual probability distribution of the ensemble members is allowed to be different from the true one. An approximation that leads to a practicable analysis algorithm is proposed. The new filter is studied in numerical experiments with a doubly stochastic onevariable model of “truth”. The model permits the assessment of the variance of the truth and the true filtering error variance at each time instance. The HBEF is shown to outperform the EnKF and the HEnKF by Myrseth and Omre (2010) in a wide range of filtering regimes in terms of performance of its primary and secondary filters.
PubDate: 20161120T14:57:33Z
DOI: 10.1016/j.physd.2016.07.009
Issue No: Vol. 338 (2016)
 Authors: Michael Tsyrulnikov; Alexander Rakitko
 The tennis racket effect in a threedimensional rigid body
 Authors: Léo Van Damme; Pavao Mardešić; Dominique Sugny
Pages: 17  25
Abstract: Publication date: 1 January 2017
Source:Physica D: Nonlinear Phenomena, Volume 338
Author(s): Léo Van Damme, Pavao Mardešić, Dominique Sugny
We propose a complete theoretical description of the tennis racket effect, which occurs in the free rotation of a threedimensional rigid body. This effect is characterized by a flip ( π  rotation) of the head of the racket when a full ( 2 π ) rotation around the unstable inertia axis is considered. We describe the asymptotics of the phenomenon and conclude about the robustness of this effect with respect to the values of the moments of inertia and the initial conditions of the dynamics. This shows the generality of this geometric property which can be found in a variety of rigid bodies. A simple analytical formula is derived to estimate the twisting effect in the general case. Different examples are discussed.
PubDate: 20161120T14:57:33Z
DOI: 10.1016/j.physd.2016.07.010
Issue No: Vol. 338 (2016)
 Authors: Léo Van Damme; Pavao Mardešić; Dominique Sugny
 Stability analysis of amplitude death in delaycoupled highdimensional
map networks and their design procedure Authors: Tomohiko Watanabe; Yoshiki Sugitani; Keiji Konishi; Naoyuki Hara
Pages: 26  33
Abstract: Publication date: 1 January 2017
Source:Physica D: Nonlinear Phenomena, Volume 338
Author(s): Tomohiko Watanabe, Yoshiki Sugitani, Keiji Konishi, Naoyuki Hara
The present paper studies amplitude death in highdimensional maps coupled by timedelay connections. A linear stability analysis provides several sufficient conditions for an amplitude death state to be unstable, i.e., an odd number property and its extended properties. Furthermore, necessary conditions for stability are provided. These conditions, which reduce trialanderror tasks for design, and the convex direction, which is a popular concept in the field of robust control, allow us to propose a design procedure for system parameters, such as coupling strength, connection delay, and input–output matrices, for a given network topology. These analytical results are confirmed numerically using delayed logistic maps, generalized Henon maps, and piecewise linear maps.
PubDate: 20161120T14:57:33Z
DOI: 10.1016/j.physd.2016.07.011
Issue No: Vol. 338 (2016)
 Authors: Tomohiko Watanabe; Yoshiki Sugitani; Keiji Konishi; Naoyuki Hara
 Generalized uncertainty principle and analogue of quantum gravity in
optics Authors: Maria Chiara Braidotti; Ziad H. Musslimani; Claudio Conti
Pages: 34  41
Abstract: Publication date: 1 January 2017
Source:Physica D: Nonlinear Phenomena, Volume 338
Author(s): Maria Chiara Braidotti, Ziad H. Musslimani, Claudio Conti
The design of optical systems capable of processing and manipulating ultrashort pulses and ultrafocused beams is highly challenging with far reaching fundamental technological applications. One key obstacle routinely encountered while implementing subwavelength optical schemes is how to overcome the limitations set by standard Fourier optics. A strategy to overcome these difficulties is to utilize the concept of a generalized uncertainty principle (GUP) which has been originally developed to study quantum gravity. In this paper we propose to use the concept of GUP within the framework of optics to show that the generalized Schrödinger equation describing short pulses and ultrafocused beams predicts the existence of a minimal spatial or temporal scale which in turn implies the existence of maximally localized states. Using a Gaussian wavepacket with complex phase, we derive the corresponding generalized uncertainty relation and its maximally localized states. Furthermore, we numerically show that the presence of nonlinearity helps the system to reach its maximal localization. Our results may trigger further theoretical and experimental tests for practical applications and analogues of fundamental physical theories.
PubDate: 20161120T14:57:33Z
DOI: 10.1016/j.physd.2016.08.001
Issue No: Vol. 338 (2016)
 Authors: Maria Chiara Braidotti; Ziad H. Musslimani; Claudio Conti
 Initial–boundary layer associated with the nonlinear
Darcy–Brinkman–Oberbeck–Boussinesq system Authors: Mingwen Fei; Daozhi Han; Xiaoming Wang
Pages: 42  56
Abstract: Publication date: 1 January 2017
Source:Physica D: Nonlinear Phenomena, Volume 338
Author(s): Mingwen Fei, Daozhi Han, Xiaoming Wang
In this paper, we study the vanishing Darcy number limit of the nonlinear Darcy–Brinkman–Oberbeck–Boussinesq system (DBOB). This singular perturbation problem involves singular structures both in time and in space giving rise to initial layers, boundary layers and initial–boundary layers. We construct an approximate solution to the DBOB system by the method of multiple scale expansions. The convergence with optimal convergence rates in certain Sobolev norms is established rigorously via the energy method.
PubDate: 20161120T14:57:33Z
DOI: 10.1016/j.physd.2016.08.002
Issue No: Vol. 338 (2016)
 Authors: Mingwen Fei; Daozhi Han; Xiaoming Wang
 Twodimensional localized structures in harmonically forced oscillatory
systems Authors: Y.P. Ma; E. Knobloch
Pages: 1  17
Abstract: Publication date: 15 December 2016
Source:Physica D: Nonlinear Phenomena, Volume 337
Author(s): Y.P. Ma, E. Knobloch
Twodimensional spatially localized structures in the complex Ginzburg–Landau equation with 1:1 resonance are studied near the simultaneous presence of a steady front between two spatially homogeneous equilibria and a supercritical Turing bifurcation on one of them. The bifurcation structures of steady circular fronts and localized target patterns are computed in the Turingstable and Turingunstable regimes. In particular, localized target patterns grow along the solution branch via ring insertion at the core in a process reminiscent of defectmediated snaking in one spatial dimension. Stability of axisymmetric solutions on these branches with respect to axisymmetric and nonaxisymmetric perturbations is determined, and parameter regimes with stable axisymmetric oscillons are identified. Direct numerical simulations reveal novel depinning dynamics of localized target patterns in the radial direction, and of circular and planar localized hexagonal patterns in the fully twodimensional system.
PubDate: 20161030T22:02:28Z
DOI: 10.1016/j.physd.2016.07.003
Issue No: Vol. 337 (2016)
 Authors: Y.P. Ma; E. Knobloch
 A comparison of macroscopic models describing the collective response of
sedimenting rodlike particles in shear flows Authors: Christiane Helzel; Athanasios E. Tzavaras
Pages: 18  29
Abstract: Publication date: 15 December 2016
Source:Physica D: Nonlinear Phenomena, Volume 337
Author(s): Christiane Helzel, Athanasios E. Tzavaras
We consider a kinetic model, which describes the sedimentation of rodlike particles in dilute suspensions under the influence of gravity, presented in Helzel and Tzavaras (submitted for publication). Here we restrict our considerations to shear flow and consider a simplified situation, where the particle orientation is restricted to the plane spanned by the direction of shear and the direction of gravity. For this simplified kinetic model we carry out a linear stability analysis and we derive two different nonlinear macroscopic models which describe the formation of clusters of higher particle density. One of these macroscopic models is based on a diffusive scaling, the other one is based on a socalled quasidynamic approximation. Numerical computations, which compare the predictions of the macroscopic models with the kinetic model, complete our presentation.
PubDate: 20161030T22:02:28Z
DOI: 10.1016/j.physd.2016.07.004
Issue No: Vol. 337 (2016)
 Authors: Christiane Helzel; Athanasios E. Tzavaras
 Exploiting stiffness nonlinearities to improve flow energy capture from
the wake of a bluff body Authors: Ali H. Alhadidi; Hamid Abderrahmane; Mohammed F. Daqaq
Pages: 30  42
Abstract: Publication date: 15 December 2016
Source:Physica D: Nonlinear Phenomena, Volume 337
Author(s): Ali H. Alhadidi, Hamid Abderrahmane, Mohammed F. Daqaq
Fluid–structure coupling mechanisms such as wake galloping have been recently utilized to develop scalable flow energy harvesters. Unlike traditional rotarytype generators which are known to suffer serious scalability issues because their efficiency drops significantly as their size decreases; wakegalloping flow energy harvesters (FEHs) operate using a very simple motion mechanism, and, hence can be scaled down to fit the desired application. Nevertheless, wakegalloping FEHs have their own shortcomings. Typically, a wakegalloping FEH has a linear restoring force which results in a very narrow lockin region. As a result, it does not perform well under the broad range of shedding frequencies normally associated with a variable flow speed. To overcome this critical problem, this article demonstrates theoretically and experimentally that, a bistable restoring force can be used to broaden the steadystate bandwidth of wake galloping FEHs and, thereby to decrease their sensitivity to variations in the flow speed. An experimental case study is carried out in a wind tunnel to compare the performance of a bistable and a linear FEH under single and multifrequency vortex street. An experimentallyvalidated lumpedparameters model of the bistable harvester is also introduced, and solved using the method of multiple scales to study the influence of the shape of the potential energy function on the output voltage.
PubDate: 20161030T22:02:28Z
DOI: 10.1016/j.physd.2016.07.005
Issue No: Vol. 337 (2016)
 Authors: Ali H. Alhadidi; Hamid Abderrahmane; Mohammed F. Daqaq
 Variety of strange pseudohyperbolic attractors in threedimensional
generalized Hénon maps Authors: A.S. Gonchenko; S.V. Gonchenko
Pages: 43  57
Abstract: Publication date: 15 December 2016
Source:Physica D: Nonlinear Phenomena, Volume 337
Author(s): A.S. Gonchenko, S.V. Gonchenko
In the present paper we focus on the problem of the existence of strange pseudohyperbolic attractors for threedimensional diffeomorphisms. Such attractors are genuine strange attractors in that sense that each orbit in the attractor has positive maximal Lyapunov exponent and this property is robust, i.e., it holds for all close systems. We restrict attention to the study of pseudohyperbolic attractors that contain only one fixed point. Then we show that threedimensional maps may have only 5 different types of such attractors, which we call the discrete Lorenz, figure8, doublefigure8, superfigure8, and superLorenz attractors. We find the first four types of attractors in threedimensional generalized Hénon maps of form x ̄ = y , y ̄ = z , z ̄ = B x + A z + C y + g ( y , z ) , where A , B and C are parameters ( B is the Jacobian) and g ( 0 , 0 ) = g ′ ( 0 , 0 ) = 0 .
PubDate: 20161030T22:02:28Z
DOI: 10.1016/j.physd.2016.07.006
Issue No: Vol. 337 (2016)
 Authors: A.S. Gonchenko; S.V. Gonchenko
 Oscillatory instabilities of gap solitons in a repulsive
Bose–Einstein condensate Authors: P.P. Kizin; D.A. Zezyulin; G.L. Alfimov
Pages: 58  66
Abstract: Publication date: 15 December 2016
Source:Physica D: Nonlinear Phenomena, Volume 337
Author(s): P.P. Kizin, D.A. Zezyulin, G.L. Alfimov
The paper is devoted to numerical study of stability of nonlinear localized modes (“gap solitons”) for the spatially onedimensional Gross–Pitaevskii equation (1D GPE) with periodic potential and repulsive interparticle interactions. We use the Evans function approach combined with the exterior algebra formulation in order to detect and describe weak oscillatory instabilities. We show that the simplest (“fundamental”) gap solitons in the first and in the second spectral gap undergo oscillatory instabilities for certain values of the frequency parameter (i.e., the chemical potential). The number of unstable eigenvalues and the associated instability rates are described. Several stable and unstable more complex (nonfundamental) gap solitons are also discussed. The results obtained from the Evans function approach are independently confirmed using the direct numerical integration of the GPE.
PubDate: 20161030T22:02:28Z
DOI: 10.1016/j.physd.2016.07.007
Issue No: Vol. 337 (2016)
 Authors: P.P. Kizin; D.A. Zezyulin; G.L. Alfimov
 Limit cycles in planar piecewise linear differential systems with
nonregular separation line Authors: Pedro Toniol Cardin; Joan Torregrosa
Pages: 67  82
Abstract: Publication date: 15 December 2016
Source:Physica D: Nonlinear Phenomena, Volume 337
Author(s): Pedro Toniol Cardin, Joan Torregrosa
In this paper we deal with planar piecewise linear differential systems defined in two zones. We consider the case when the two linear zones are angular sectors of angles α and 2 π − α , respectively, for α ∈ ( 0 , π ) . We study the problem of determining lower bounds for the number of isolated periodic orbits in such systems using Melnikov functions. These limit cycles appear studying higher order piecewise linear perturbations of a linear center. It is proved that the maximum number of limit cycles that can appear up to a sixth order perturbation is five. Moreover, for these values of α , we prove the existence of systems with four limit cycles up to fifth order and, for α = π / 2 , we provide an explicit example with five up to sixth order. In general, the nonregular separation line increases the number of periodic orbits in comparison with the case where the two zones are separated by a straight line.
PubDate: 20161030T22:02:28Z
DOI: 10.1016/j.physd.2016.07.008
Issue No: Vol. 337 (2016)
 Authors: Pedro Toniol Cardin; Joan Torregrosa
 Causal hydrodynamics from kinetic theory by doublet scheme in
renormalizationgroup method Authors: Kyosuke Tsumura; Yuta Kikuchi; Teiji Kunihiro
Pages: 1  27
Abstract: Publication date: 1 December 2016
Source:Physica D: Nonlinear Phenomena, Volume 336
Author(s): Kyosuke Tsumura, Yuta Kikuchi, Teiji Kunihiro
We develop a general framework in the renormalizationgroup (RG) method for extracting a mesoscopic dynamics from an evolution equation by incorporating some excited (fast) modes as additional components to the invariant manifold spanned by zero modes. We call this framework the doublet scheme. The validity of the doublet scheme is first tested and demonstrated by taking the Lorenz model as a simple threedimensional dynamical system; it is shown that the twodimensional reduced dynamics on the attractive manifold composed of the wouldbe zero and a fast modes are successfully obtained in a natural way. We then apply the doublet scheme to construct causal hydrodynamics as a mesoscopic dynamics of kinetic theory, i.e., the Boltzmann equation, in a systematic manner with no adhoc assumption. It is found that our equation has the same form as Grad’s thirteenmoment causal hydrodynamic equation, but the microscopic formulae of the transport coefficients and relaxation times are different. In fact, in contrast to the Grad equation, our equation leads to the same expressions for the transport coefficients as given by the Chapman–Enskog expansion method and suggests novel formulae of the relaxation times expressed in terms of relaxation functions which allow a natural physical interpretation of the relaxation times. Furthermore, our theory nicely gives the explicit forms of the distribution function and the thirteen hydrodynamic variables in terms of the linearized collision operator, which in turn clearly suggest the proper ansatz forms of them to be adopted in the method of moments.
PubDate: 20161016T12:47:09Z
DOI: 10.1016/j.physd.2016.06.012
Issue No: Vol. 336 (2016)
 Authors: Kyosuke Tsumura; Yuta Kikuchi; Teiji Kunihiro
 Towards the modeling of nanoindentation of virus shells: Do substrate
adhesion and geometry matter? Authors: Arthur Bousquet; Bogdan Dragnea; Manel Tayachi; Roger Temam
Pages: 28  38
Abstract: Publication date: 1 December 2016
Source:Physica D: Nonlinear Phenomena, Volume 336
Author(s): Arthur Bousquet, Bogdan Dragnea, Manel Tayachi, Roger Temam
Soft nanoparticles adsorbing at surfaces undergo deformation and buildup of elastic strain as a consequence of interfacial adhesion of similar magnitude with constitutive interactions. An example is the adsorption of virus particles at surfaces, a phenomenon of central importance for experiments in virus nanoindentation and for understanding of virus entry. The influence of adhesion forces and substrate corrugation on the mechanical response to indentation has not been studied. This is somewhat surprising considering that many singlestranded RNA icosahedral viruses are organized by soft intermolecular interactions while relatively strong adhesion forces are required for virus immobilization for nanoindentation. This article presents numerical simulations via finite elements discretization investigating the deformation of a thick shell in the context of slow evolution linear elasticity and in presence of adhesion interactions with the substrate. We study the influence of the adhesion forces in the deformation of the virus model under axial compression on a flat substrate by comparing the force–displacement curves for a shell having elastic constants relevant to virus capsids with and without adhesion forces derived from the LennardJones potential. Finally, we study the influence of the geometry of the substrate in twodimensions by comparing deformation of the virus model adsorbed at the cusp between two cylinders with that on a flat surface.
PubDate: 20161016T12:47:09Z
DOI: 10.1016/j.physd.2016.06.013
Issue No: Vol. 336 (2016)
 Authors: Arthur Bousquet; Bogdan Dragnea; Manel Tayachi; Roger Temam
 Fluctuations induced transition of localization of granular objects caused
by degrees of crowding Authors: Soutaro Oda; Yoshitsugu Kubo; ChwenYang Shew; Kenichi Yoshikawa
Pages: 39  46
Abstract: Publication date: 1 December 2016
Source:Physica D: Nonlinear Phenomena, Volume 336
Author(s): Soutaro Oda, Yoshitsugu Kubo, ChwenYang Shew, Kenichi Yoshikawa
Fluctuations are ubiquitous in both microscopic and macroscopic systems, and an investigation of confined particles under fluctuations is relevant to how living cells on the earth maintain their lives. Inspired by biological cells, we conduct the experiment through a very simple fluctuating system containing one or several large spherical granular particles and multiple smaller ones confined on a cylindrical dish under vertical vibration. We find a universal behavior that large particles preferentially locate in cavity interior due to the fact that large particles are depleted from the cavity wall by small spheres under vertical vibration in the actual experiment. This universal behavior can be understood from the standpoint of entropy.
PubDate: 20161016T12:47:09Z
DOI: 10.1016/j.physd.2016.06.014
Issue No: Vol. 336 (2016)
 Authors: Soutaro Oda; Yoshitsugu Kubo; ChwenYang Shew; Kenichi Yoshikawa
 Coupled oscillators on evolving networks
 Authors: R.K. Singh; Trilochan Bagarti
Pages: 47  52
Abstract: Publication date: 1 December 2016
Source:Physica D: Nonlinear Phenomena, Volume 336
Author(s): R.K. Singh, Trilochan Bagarti
In this work we study coupled oscillators on evolving networks. We find that the steady state behavior of the system is governed by the relative values of the spread in natural frequencies and the global coupling strength. For coupling strong in comparison to the spread in frequencies, the system of oscillators synchronize and when coupling strength and spread in frequencies are large, a phenomenon similar to amplitude death is observed. The network evolution provides a mechanism to build interoscillator connections and once a dynamic equilibrium is achieved, oscillators evolve according to their local interactions. We also find that the steady state properties change by the presence of additional time scales. We demonstrate these results based on numerical calculations studying dynamical evolution of limitcycle and van der Pol oscillators.
PubDate: 20161016T12:47:09Z
DOI: 10.1016/j.physd.2016.06.015
Issue No: Vol. 336 (2016)
 Authors: R.K. Singh; Trilochan Bagarti
 Emergence of chaos in a spatially confined reactive system
 Authors: Valérie Voorsluijs; Yannick De Decker
Pages: 1  9
Abstract: Publication date: 15 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 335
Author(s): Valérie Voorsluijs, Yannick De Decker
In spatially restricted media, interactions between particles and local fluctuations of density can lead to important deviations of the dynamics from the unconfined, deterministic picture. In this context, we investigated how molecular crowding can affect the emergence of chaos in small reactive systems. We developed to this end an amended version of the Willamowski–Rössler model, where we account for the impenetrability of the reactive species. We analyzed the deterministic kinetics of this model and studied it with spatiallyextended stochastic simulations in which the mobility of particles is included explicitly. We show that homogeneous fluctuations can lead to a destruction of chaos through a fluctuationinduced collision between chaotic trajectories and absorbing states. However, an interplay between the size of the system and the mobility of particles can counterbalance this effect so that chaos can indeed be found when particles diffuse slowly. This unexpected effect can be traced back to the emergence of spatial correlations which strongly affect the dynamics. The mobility of particles effectively acts as a new bifurcation parameter, enabling the system to switch from stationary states to absorbing states, oscillations or chaos.
PubDate: 20161002T03:34:48Z
DOI: 10.1016/j.physd.2016.05.005
Issue No: Vol. 335 (2016)
 Authors: Valérie Voorsluijs; Yannick De Decker
 Breather solutions for inhomogeneous FPU models using Birkhoff normal
forms Authors: Francisco MartínezFarías; Panayotis Panayotaros
Pages: 10  25
Abstract: Publication date: 15 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 335
Author(s): Francisco MartínezFarías, Panayotis Panayotaros
We present results on spatially localized oscillations in some inhomogeneous nonlinear lattices of Fermi–Pasta–Ulam (FPU) type derived from phenomenological nonlinear elastic network models proposed to study localized protein vibrations. The main feature of the FPU lattices we consider is that the number of interacting neighbors varies from site to site, and we see numerically that this spatial inhomogeneity leads to spatially localized normal modes in the linearized problem. This property is seen in 1D models, and in a 3D model with a geometry obtained from protein data. The spectral analysis of these examples suggests some nonresonance assumptions that we use to show the existence of invariant subspaces of spatially localized solutions in quartic Birkhoff normal forms of the FPU systems. The invariant subspaces have an additional symmetry and this fact allows us to compute periodic orbits of the quartic normal form in a relatively simple way.
PubDate: 20161002T03:34:48Z
DOI: 10.1016/j.physd.2016.06.004
Issue No: Vol. 335 (2016)
 Authors: Francisco MartínezFarías; Panayotis Panayotaros
 Kinetic theory of cluster dynamics
 Authors: Robert I.A. Patterson; Sergio Simonella; Wolfgang Wagner
Pages: 26  32
Abstract: Publication date: 15 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 335
Author(s): Robert I.A. Patterson, Sergio Simonella, Wolfgang Wagner
In a Newtonian system with localized interactions the whole set of particles is naturally decomposed into dynamical clusters, defined as finite groups of particles having an influence on each other’s trajectory during a given interval of time. For an ideal gas with shortrange intermolecular force, we provide a description of the cluster size distribution in terms of the reduced Boltzmann density. In the simplified context of Maxwell molecules, we show that a macroscopic fraction of the gas forms a giant component in finite kinetic time. The critical index of this phase transition is in agreement with previous numerical results on the elastic billiard.
PubDate: 20161002T03:34:48Z
DOI: 10.1016/j.physd.2016.06.007
Issue No: Vol. 335 (2016)
 Authors: Robert I.A. Patterson; Sergio Simonella; Wolfgang Wagner
 Frequency locking near the gluing bifurcation: Spintorque oscillator
under periodic modulation of current Authors: Michael A. Zaks; Arkady Pikovsky
Pages: 33  44
Abstract: Publication date: 15 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 335
Author(s): Michael A. Zaks, Arkady Pikovsky
We consider entrainment by periodic force of limit cycles which are close to the homoclinic bifurcation. Taking as a physical example the nanoscale spintorque oscillator in the LC circuit, we develop the general description of the situation in which the frequency of the stable periodic orbit in the autonomous system is highly sensitive to minor variations of the parameter, and derive explicit expressions for the strongly deformed borders of the resonance regions (Arnold tongues) in the parameter space of the problem. It turns out that proximity to homoclinic bifurcations hinders synchronization of spintorque oscillators.
PubDate: 20161002T03:34:48Z
DOI: 10.1016/j.physd.2016.06.008
Issue No: Vol. 335 (2016)
 Authors: Michael A. Zaks; Arkady Pikovsky
 Chaotic subdynamics in coupled logistic maps
 Authors: Marek Lampart; Piotr Oprocha
Pages: 45  53
Abstract: Publication date: 15 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 335
Author(s): Marek Lampart, Piotr Oprocha
We study the dynamics of Laplaciantype coupling induced by logistic family f μ ( x ) = μ x ( 1 − x ) , where μ ∈ [ 0 , 4 ] , on a periodic lattice, that is the dynamics of maps of the form F ( x , y ) = ( ( 1 − ε ) f μ ( x ) + ε f μ ( y ) , ( 1 − ε ) f μ ( y ) + ε f μ ( x ) ) where ε > 0 determines strength of coupling. Our main objective is to analyze the structure of attractors in such systems and especially detect invariant regions with nontrivial dynamics outside the diagonal. In analytical way, we detect some regions of parameters for which a horseshoe is present; and using simulations global attractors and invariant sets are depicted.
PubDate: 20161002T03:34:48Z
DOI: 10.1016/j.physd.2016.06.010
Issue No: Vol. 335 (2016)
 Authors: Marek Lampart; Piotr Oprocha
 A hierarchy of Poisson brackets in nonequilibrium thermodynamics
 Authors: Michal Pavelka; Václav Klika; Oğul Esen; Miroslav Grmela
Pages: 54  69
Abstract: Publication date: 15 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 335
Author(s): Michal Pavelka, Václav Klika, Oğul Esen, Miroslav Grmela
Reversible evolution of macroscopic and mesoscopic systems can be conveniently constructed from two ingredients: an energy functional and a Poisson bracket. The goal of this paper is to elucidate how the Poisson brackets can be constructed and what additional features we also gain by the construction. In particular, the Poisson brackets governing reversible evolution in oneparticle kinetic theory, kinetic theory of binary mixtures, binary fluid mixtures, classical irreversible thermodynamics and classical hydrodynamics are derived from Liouville equation. Although the construction is quite natural, a few examples where it does not work are included (e.g. the BBGKY hierarchy). Finally, a new infinite grandcanonical hierarchy of Poisson brackets is proposed, which leads to Poisson brackets expressing nonlocal phenomena such as turbulent motion or evolution of polymeric fluids. Eventually, Lie–Poisson structures standing behind some of the brackets are identified.
PubDate: 20161002T03:34:48Z
DOI: 10.1016/j.physd.2016.06.011
Issue No: Vol. 335 (2016)
 Authors: Michal Pavelka; Václav Klika; Oğul Esen; Miroslav Grmela
 Topology in Dynamics, Differential Equations, and Data
 Authors: Sarah Day; Robertus C.A.M. Vandervorst; Thomas Wanner
Pages: 1  3
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Sarah Day, Robertus C.A.M. Vandervorst, Thomas Wanner
This special issue is devoted to showcasing recent uses of topological methods in the study of dynamical behavior and the analysis of both numerical and experimental data. The twelve original research papers span a wide spectrum of results from abstract index theories, over homology and persistencebased data analysis techniques, to computerassisted proof techniques based on topological fixed point arguments.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.08.003
Issue No: Vol. 334 (2016)
 Authors: Sarah Day; Robertus C.A.M. Vandervorst; Thomas Wanner
 Geometric phase in the Hopf bundle and the stability of nonlinear waves
 Authors: Colin J. Grudzien; Thomas J. Bridges; Christopher K.R.T. Jones
Pages: 4  18
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Colin J. Grudzien, Thomas J. Bridges, Christopher K.R.T. Jones
We develop a stability index for the traveling waves of nonlinear reaction–diffusion equations using the geometric phase induced on the Hopf bundle S 2 n − 1 ⊂ C n . This can be viewed as an alternative formulation of the winding number calculation of the Evans function, whose zeros correspond to the eigenvalues of the linearization of reaction–diffusion operators about the wave. The stability of a traveling wave can be determined by the existence of eigenvalues of positive real part for the linear operator. Our method of geometric phase for locating and counting eigenvalues is inspired by the numerical results in Way’s Dynamics in the Hopf bundle, the geometric phase and implications for dynamical systems Way (2009). We provide a detailed proof of the relationship between the phase and eigenvalues for dynamical systems defined on C 2 and sketch the proof of the method of geometric phase for C n and its generalization to boundaryvalue problems. Implementing the numerical method, modified from Way (2009), we conclude with open questions inspired from the results.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.04.005
Issue No: Vol. 334 (2016)
 Authors: Colin J. Grudzien; Thomas J. Bridges; Christopher K.R.T. Jones
 The Poincaré–Bendixson Theorem and the nonlinear
Cauchy–Riemann equations Authors: J.B. van den Berg; S. Munaò; R.C.A.M. Vandervorst
Pages: 19  28
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): J.B. van den Berg, S. Munaò, R.C.A.M. Vandervorst
Fiedler and MalletParet (1989) prove a version of the classical Poincaré–Bendixson Theorem for scalar parabolic equations. We prove that a similar result holds for bounded solutions of the nonlinear Cauchy–Riemann equations. The latter is an application of an abstract theorem for flows with a(n) (unbounded) discrete Lyapunov function.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.04.009
Issue No: Vol. 334 (2016)
 Authors: J.B. van den Berg; S. Munaò; R.C.A.M. Vandervorst
 Arnold’s mechanism of diffusion in the spatial circular restricted
threebody problem: A semianalytical argument Authors: Amadeu Delshams; Marian Gidea; Pablo Roldan
Pages: 29  48
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Amadeu Delshams, Marian Gidea, Pablo Roldan
We consider the spatial circular restricted threebody problem, on the motion of an infinitesimal body under the gravity of Sun and Earth. This can be described by a 3degree of freedom Hamiltonian system. We fix an energy level close to that of the collinear libration point L 1 , located between Sun and Earth. Near L 1 there exists a normally hyperbolic invariant manifold, diffeomorphic to a 3sphere. For an orbit confined to this 3sphere, the amplitude of the motion relative to the ecliptic (the plane of the orbits of Sun and Earth) can vary only slightly. We show that we can obtain new orbits whose amplitude of motion relative to the ecliptic changes significantly, by following orbits of the flow restricted to the 3sphere alternatively with homoclinic orbits that turn around the Earth. We provide an abstract theorem for the existence of such ‘diffusing’ orbits, and numerical evidence that the premises of the theorem are satisfied in the threebody problem considered here. We provide an explicit construction of diffusing orbits. The geometric mechanism underlying this construction is reminiscent of the Arnold diffusion problem for Hamiltonian systems. Our argument, however, does not involve transition chains of tori as in the classical example of Arnold. We exploit mostly the ‘outer dynamics’ along homoclinic orbits, and use very little information on the ‘inner dynamics’ restricted to the 3sphere. As a possible application to astrodynamics, diffusing orbits as above can be used to design low cost maneuvers to change the inclination of an orbit of a satellite near L 1 from a nearlyplanar orbit to a tilted orbit with respect to the ecliptic. We explore different energy levels, and estimate the largest orbital inclination that can be achieved through our construction.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.06.005
Issue No: Vol. 334 (2016)
 Authors: Amadeu Delshams; Marian Gidea; Pablo Roldan
 Exploring the topology of dynamical reconstructions
 Authors: Joshua Garland; Elizabeth Bradley; James D. Meiss
Pages: 49  59
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Joshua Garland, Elizabeth Bradley, James D. Meiss
Computing the statespace topology of a dynamical system from scalar data requires accurate reconstruction of those dynamics and construction of an appropriate simplicial complex from the results. The reconstruction process involves a number of free parameters and the computation of homology for a large number of simplices can be expensive. This paper is a study of how to compute the homology efficiently and effectively without a full (diffeomorphic) reconstruction. Using trajectories from the classic Lorenz system, we reconstruct the dynamics using the method of delays, then build a simplicial complex whose vertices are a small subset of the data: the “witness complex”. Surprisingly, we find that the witness complex correctly resolves the homology of the underlying invariant set from noisy samples of that set even if the reconstruction dimension is well below the thresholds for assuring topological conjugacy between the true and reconstructed dynamics that are specified in the embedding theorems. We conjecture that this is because the requirements for reconstructing homology are less stringent: a homeomorphism is sufficient—as opposed to a diffeomorphism, as is necessary for the full dynamics. We provide preliminary evidence that a homeomorphism, in the form of a delaycoordinate reconstruction map, may exist at a lower dimension than that required to achieve an embedding.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.03.006
Issue No: Vol. 334 (2016)
 Authors: Joshua Garland; Elizabeth Bradley; James D. Meiss
 Topological microstructure analysis using persistence landscapes
 Authors: Paweł Dłotko; Thomas Wanner
Pages: 60  81
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Paweł Dłotko, Thomas Wanner
Phase separation mechanisms can produce a variety of complicated and intricate microstructures, which often can be difficult to characterize in a quantitative way. In recent years, a number of novel topological metrics for microstructures have been proposed, which measure essential connectivity information and are based on techniques from algebraic topology. Such metrics are inherently computable using computational homology, provided the microstructures are discretized using a thresholding process. However, while in many cases the thresholding is straightforward, noise and measurement errors can lead to misleading metric values. In such situations, persistence landscapes have been proposed as a natural topology metric. Common to all of these approaches is the enormous data reduction, which passes from complicated patterns to discrete information. It is therefore natural to wonder what type of information is actually retained by the topology. In the present paper, we demonstrate that averaged persistence landscapes can be used to recover central system information in the Cahn–Hilliard theory of phase separation. More precisely, we show that topological information of evolving microstructures alone suffices to accurately detect both concentration information and the actual decomposition stage of a data snapshot. Considering that persistent homology only measures discrete connectivity information, regardless of the size of the topological features, these results indicate that the system parameters in a phase separation process affect the topology considerably more than anticipated. We believe that the methods discussed in this paper could provide a valuable tool for relating experimental data to model simulations.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.04.015
Issue No: Vol. 334 (2016)
 Authors: Paweł Dłotko; Thomas Wanner
 Analysis of Kolmogorov flow and Rayleigh–Bénard convection
using persistent homology Authors: Miroslav Kramár; Rachel Levanger; Jeffrey Tithof; Balachandra Suri; Mu Xu; Mark Paul; Michael F. Schatz; Konstantin Mischaikow
Pages: 82  98
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Miroslav Kramár, Rachel Levanger, Jeffrey Tithof, Balachandra Suri, Mu Xu, Mark Paul, Michael F. Schatz, Konstantin Mischaikow
We use persistent homology to build a quantitative understanding of large complex systems that are driven farfromequilibrium. In particular, we analyze image time series of flow field patterns from numerical simulations of two important problems in fluid dynamics: Kolmogorov flow and Rayleigh–Bénard convection. For each image we compute a persistence diagram to yield a reduced description of the flow field; by applying different metrics to the space of persistence diagrams, we relate characteristic features in persistence diagrams to the geometry of the corresponding flow patterns. We also examine the dynamics of the flow patterns by a second application of persistent homology to the time series of persistence diagrams. We demonstrate that persistent homology provides an effective method both for quotienting out symmetries in families of solutions and for identifying multiscale recurrent dynamics. Our approach is quite general and it is anticipated to be applicable to a broad range of open problems exhibiting complex spatiotemporal behavior.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.02.003
Issue No: Vol. 334 (2016)
 Authors: Miroslav Kramár; Rachel Levanger; Jeffrey Tithof; Balachandra Suri; Mu Xu; Mark Paul; Michael F. Schatz; Konstantin Mischaikow
 Principal component analysis of persistent homology rank functions with
case studies of spatial point patterns, sphere packing and colloids Authors: Vanessa Robins; Katharine Turner
Pages: 99  117
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Vanessa Robins, Katharine Turner
Persistent homology, while ostensibly measuring changes in topology, captures multiscale geometrical information. It is a natural tool for the analysis of point patterns. In this paper we explore the statistical power of the persistent homology rank functions. For a point pattern X we construct a filtration of spaces by taking the union of balls of radius a centred on points in X , X a = ∪ x ∈ X B ( x , a ) . The rank function β k ( X ) : { ( a , b ) ∈ R 2 : a ≤ b } → R is then defined by β k ( X ) ( a , b ) = rank ( ι ∗ : H k ( X a ) → H k ( X b ) ) where ι ∗ is the induced map on homology from the inclusion map on spaces. We consider the rank functions as lying in a Hilbert space and show that under reasonable conditions the rank functions from multiple simulations or experiments will lie in an affine subspace. This enables us to perform functional principal component analysis which we apply to experimental data from colloids at different effective temperatures and to sphere packings with different volume fractions. We also investigate the potential of rank functions in providing a test of complete spatial randomness of 2D point patterns using the distances to an empirically computed mean rank function of binomial point patterns in the unit square.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.03.007
Issue No: Vol. 334 (2016)
 Authors: Vanessa Robins; Katharine Turner
 Continuation of point clouds via persistence diagrams
 Authors: Marcio Gameiro; Yasuaki Hiraoka; Ippei Obayashi
Pages: 118  132
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Marcio Gameiro, Yasuaki Hiraoka, Ippei Obayashi
In this paper, we present a mathematical and algorithmic framework for the continuation of point clouds by persistence diagrams. A key property used in the method is that the persistence map, which assigns a persistence diagram to a point cloud, is differentiable. This allows us to apply the Newton–Raphson continuation method in this setting. Given an original point cloud P , its persistence diagram D , and a target persistence diagram D ′ , we gradually move from D to D ′ , by successively computing intermediate point clouds until we finally find a point cloud P ′ having D ′ as its persistence diagram. Our method can be applied to a wide variety of situations in topological data analysis where it is necessary to solve an inverse problem, from persistence diagrams to point cloud data.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2015.11.011
Issue No: Vol. 334 (2016)
 Authors: Marcio Gameiro; Yasuaki Hiraoka; Ippei Obayashi
 Chaos near a resonant inclinationflip
 Authors: Marcus Fontaine; William Kalies; Vincent Naudot
Pages: 141  157
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Marcus Fontaine, William Kalies, Vincent Naudot
Horseshoes play a central role in dynamical systems and are observed in many chaotic systems. However most points in a neighborhood of the horseshoe escape after finitely many iterations. In this work we construct a new model by reinjecting the points that escape the horseshoe. We show that this model can be realized within an attractor of a flow arising from a threedimensional vector field, after perturbation of an inclinationflip homoclinic orbit with a resonance. The dynamics of this model, without considering the reinjection, often contains a cuspidal horseshoe with positive entropy, and we show that for a computational example the dynamics with reinjection can have more complexity than the cuspidal horseshoe alone.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.06.009
Issue No: Vol. 334 (2016)
 Authors: Marcus Fontaine; William Kalies; Vincent Naudot
 Rigorous numerics for NLS: Bound states, spectra, and controllability
 Authors: Roberto Castelli; Holger Teismann
Pages: 158  173
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Roberto Castelli, Holger Teismann
In this paper it is demonstrated how rigorous numerics may be applied to the onedimensional nonlinear Schrödinger equation (NLS); specifically, to determining boundstate solutions and establishing certain spectral properties of the linearization. Since the results are rigorous, they can be used to complete a recent analytical proof (Beauchard et al., 2015) of the local exact controllability of NLS.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.01.005
Issue No: Vol. 334 (2016)
 Authors: Roberto Castelli; Holger Teismann
 Automatic differentiation for Fourier series and the radii polynomial
approach Authors: JeanPhilippe Lessard; J.D. Mireles James; Julian Ransford
Pages: 174  186
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): JeanPhilippe Lessard, J.D. Mireles James, Julian Ransford
In this work we develop a computerassisted technique for proving existence of periodic solutions of nonlinear differential equations with nonpolynomial nonlinearities. We exploit ideas from the theory of automatic differentiation in order to formulate an augmented polynomial system. We compute a numerical Fourier expansion of the periodic orbit for the augmented system, and prove the existence of a true solution nearby using an aposteriori validation scheme (the radii polynomial approach). The problems considered here are given in terms of locally analytic vector fields (i.e. the field is analytic in a neighborhood of the periodic orbit) hence the computerassisted proofs are formulated in a Banach space of sequences satisfying a geometric decay condition. In order to illustrate the use and utility of these ideas we implement a number of computerassisted existence proofs for periodic orbits of the Planar Circular Restricted ThreeBody Problem (PCRTBP).
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.02.007
Issue No: Vol. 334 (2016)
 Authors: JeanPhilippe Lessard; J.D. Mireles James; Julian Ransford
 Excitability, mixedmode oscillations and transition to chaos in a
stochastic ice ages model Authors: D.V. Alexandrov; I.A. Bashkirtseva; L.B. Ryashko
Abstract: Publication date: Available online 30 November 2016
Source:Physica D: Nonlinear Phenomena
Author(s): D.V. Alexandrov, I.A. Bashkirtseva, L.B. Ryashko
Motivated by an important geophysical significance, we consider the influence of stochastic forcing on a simple threedimensional climate model previously derived by Saltzman and Sutera. A nonlinear dynamical system governing three physical variables, the bulk ocean temperature, continental and marine ice masses, is analyzed in deterministic and stochastic cases. It is shown that the attractor of deterministic model is either a stable equilibrium or a limit cycle. We demonstrate that the process of continental ice melting occurs with a noisedependent time delay as compared with marine ice melting. The paleoclimate cyclicity which is near 100 ky in a wide range of model parameters abruptly increases in the vicinity of a bifurcation point and depends on the noise intensity. In a zone of stable equilibria, the 3D climate model under consideration is extremely excitable. Even for a weak random noise, the stochastic trajectories demonstrate a transition from small to largeamplitude stochastic oscillations (SLASO). In a zone of stable cycles, SLASO transitions are analyzed too. We show that such stochastic transitions play an important role in the formation of a mixedmode paleoclimate scenario. This mixedmode dynamics with the intermittency of large and smallamplitude stochastic oscillations and coherence resonance are investigated via analysis of interspike intervals. A tendency of dynamic paleoclimate to abrupt and rapid glaciations and deglaciations as well as its transition from order to chaos with increasing noise are shown.
PubDate: 20161204T02:03:09Z
DOI: 10.1016/j.physd.2016.11.007
 Authors: D.V. Alexandrov; I.A. Bashkirtseva; L.B. Ryashko
 One and twodimensional bright solitons in inhomogeneous defocusing
nonlinearities with an antisymmetric periodic gain and loss Authors: Dengchu Guo; Jing Xiao; Linlin Gu; Hongzhen Jin; Liangwei Dong
Abstract: Publication date: Available online 2 December 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Dengchu Guo, Jing Xiao, Linlin Gu, Hongzhen Jin, Liangwei Dong
We address that various branches of bright solitons exist in a spatially inhomogeneous defocusing nonlinearity with an imprinted antisymmetric periodic gainloss profile. The spectra of such systems with a purely imaginary potential never become complex and thus the paritytime symmetry is unbreakable. The mergence between pairs of soliton branches occurs at a critical gainloss strength, above which no soliton solutions can be found. Intriguingly, which pair of soliton branches will merge together can be changed by varying the modulation frequency of gain and loss. Most branches of onedimensional solitons are stable in wide parameter regions. We also provide the first example of twodimensional bright solitons with unbreakable paritytime symmetry.
PubDate: 20161204T02:03:09Z
DOI: 10.1016/j.physd.2016.11.005
 Authors: Dengchu Guo; Jing Xiao; Linlin Gu; Hongzhen Jin; Liangwei Dong
 A nonperturbative analytic expression of signal amplification factor in
stochastic resonance Authors: Asish Kumar Dhara
Abstract: Publication date: Available online 21 November 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Asish Kumar Dhara
We put forward a nonperturbative scheme to calculate the response of an overdamped bistable system driven by a Gaussian white noise and perturbed by a weak monochromatic force (signal) analytically. The formalism takes into account infinite number of perturbation terms of a perturbation series with amplitude of the signal as an expansion parameter. The contributions of infinite number of relaxation modes of the stochastic dynamics to the response are also taken into account in this formalism. A closed form analytic expression of the response is obtained. Only the knowledge of the first nontrivial eigenvalue and the lowest eigenfunction of the unperturbed Fokker–Planck operator are needed to evaluate the response. The response calculated from the derived analytic expression matches fairly well with the numerical results.
PubDate: 20161127T16:42:30Z
DOI: 10.1016/j.physd.2016.11.002
 Authors: Asish Kumar Dhara
 Optical dispersive shock waves in defocusing colloidal media
 Authors: X. An; T.R. Marchant; N.F. Smyth
Abstract: Publication date: Available online 24 November 2016
Source:Physica D: Nonlinear Phenomena
Author(s): X. An, T.R. Marchant, N.F. Smyth
The propagation of an optical dispersive shock wave, generated from a jump discontinuity in light intensity, in a defocussing colloidal medium is analysed. The equations governing nonlinear light propagation in a colloidal medium consist of a nonlinear Schrödinger equation for the beam and an algebraic equation for the medium response. In the limit of low light intensity, these equations reduce to a perturbed higher order nonlinear Schrödinger equation. Solutions for the leading and trailing edges of the colloidal dispersive shock wave are found using modulation theory. This is done for both the perturbed nonlinear Schrödinger equation and the full colloid equations for arbitrary light intensity. These results are compared with numerical solutions of the colloid equations.
PubDate: 20161127T16:42:30Z
DOI: 10.1016/j.physd.2016.11.004
 Authors: X. An; T.R. Marchant; N.F. Smyth
 Lowdimensional reducedorder models for statistical response and
uncertainty quantification: Barotropic turbulence with topography Authors: Di Qi; Andrew J. Majda
Abstract: Publication date: Available online 25 November 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Di Qi, Andrew J. Majda
A lowdimensional reducedorder statistical closure model is developed for quantifying the uncertainty to changes in forcing in a barotropic turbulent system with topography involving interactions between smallscale motions and a largescale mean flow. Imperfect model sensitivity is improved through a recent mathematical strategy for calibrating model errors in a training phase, where information theory and linear statistical response theory are combined in a systematic fashion to achieve the optimal model parameters. Statistical theories about a Gaussian invariant measure and the exact statistical energy equations are also developed for the truncated barotropic equations that can be used to improve the imperfect model prediction skill. A stringent paradigm model of 57 degrees of freedom is used to display the feasibility of the reducedorder methods. This simple model creates largescale zonal mean flow shifting directions from westward to eastward jets with an abrupt change in amplitude when perturbations are applied, and prototype blocked and unblocked patterns can be generated in this simple model similar to the real natural system. Principal statistical responses in mean and variance can be captured by the reducedorder models with desirable accuracy and efficiency with only 3 resolved modes. An even more challenging regime with nonGaussian equilibrium statistics using the fluctuation equations is also tested in the reducedorder models with accurate prediction using the first 5 resolved modes. These reducedorder models also show potential for uncertainty quantification and prediction in more complex realistic geophysical turbulent dynamical systems.
PubDate: 20161127T16:42:30Z
DOI: 10.1016/j.physd.2016.11.006
 Authors: Di Qi; Andrew J. Majda
 Macroscopic heat transport equations and heat waves in nonequilibrium
states Authors: Yangyu Guo; David Jou; Moran Wang
Abstract: Publication date: Available online 16 November 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Yangyu Guo, David Jou, Moran Wang
Heat transport may behave as wave propagation when the time scale of processes decreases to be comparable to or smaller than the relaxation time of heat carriers. In this work, a generalized heat transport equation including nonlinear, nonlocal and relaxation terms is proposed, which sums up the CattaneoVernotte, dualphaselag and phonon hydrodynamic models as special cases. In the frame of this equation, the heat wave propagations are investigated systematically in nonequilibrium steady states, which were usually studied around equilibrium states. The phase (or front) speed of heat waves is obtained through a perturbation solution to the heat differential equation, and found to be intimately related to the nonlinear and nonlocal terms. Thus, potential heat wave experiments in nonequilibrium states are devised to measure the coefficients in the generalized equation, which may throw light on understanding the physical mechanisms and macroscopic modeling of nanoscale heat transport.
PubDate: 20161120T14:57:33Z
DOI: 10.1016/j.physd.2016.10.005
 Authors: Yangyu Guo; David Jou; Moran Wang
 Spatiotemporal control to eliminate cardiac alternans using isostable
reduction Authors: Dan Wilson; Jeff Moehlis
Abstract: Publication date: Available online 16 November 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Dan Wilson, Jeff Moehlis
Cardiac alternans, an arrhythmia characterized by a beattobeat alternation of cardiac action potential durations, is widely believed to facilitate the transition from normal cardiac function to ventricular fibrillation and sudden cardiac death. Alternans arises due to an instability of a healthy period1 rhythm, and most dynamical control strategies either require extensive knowledge of the cardiac system, making experimental validation difficult, or are model independent and sacrifice important information about the specific system under study. Isostable reduction provides an alternative approach, in which the response of a system to external perturbations can be used to reduce the complexity of a cardiac system, making it easier to work with from an analytical perspective while retaining many of its important features. Here, we use isostable reduction strategies to reduce the complexity of partial differential equation models of cardiac systems in order to develop energy optimal strategies for the elimination of alternans. Resulting control strategies require significantly less energy to terminate alternans than comparable strategies and do not require continuous state feedback.
PubDate: 20161120T14:57:33Z
DOI: 10.1016/j.physd.2016.11.001
 Authors: Dan Wilson; Jeff Moehlis
 Finitetime thin film rupture driven by modified evaporative loss
 Authors: Hangjie Ji; Thomas P. Witelski
Abstract: Publication date: Available online 31 October 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Hangjie Ji, Thomas P. Witelski
Rupture is a nonlinear instability resulting in a finitetime singularity as a film layer approaches zero thickness at a point. We study the dynamics of rupture in a generalized mathematical model of thin films of viscous fluids with modified evaporative effects. The governing lubrication model is a fourthorder nonlinear parabolic partial differential equation with a nonconservative loss term. Several different types of finitetime singularities are observed due to balances between conservative and nonconservative terms. Nonselfsimilar behavior and two classes of selfsimilar rupture solutions are analyzed and validated against high resolution PDE simulations.
PubDate: 20161106T22:10:15Z
DOI: 10.1016/j.physd.2016.10.002
 Authors: Hangjie Ji; Thomas P. Witelski
 Stability on timedependent domains: Convective and dilution effects
 Authors: R. Krechetnikov; E. Knobloch
Abstract: Publication date: Available online 28 October 2016
Source:Physica D: Nonlinear Phenomena
Author(s): R. Krechetnikov, E. Knobloch
In this paper we explore nearcritical behavior of spatially extended systems on timedependent spatial domains with convective and dilution effects due to domain flow. As a paradigm, we use the SwiftHohenberg equation, which is the simplest nonlinear model with a finite nonzero critical wavenumber, to study dynamic pattern formation on timedependent domains. A universal amplitude equation governing weakly nonlinear evolution of the pattern on timedependent domains is derived and proves to be a generalization of the standard GinzburgLandau equation. Its key solutions identified here demonstrate a substantial variety–spatially periodic states with a timedependent wavenumber, steady spatially nonperiodic states, and pulsetrain solutions–in contrast to extended systems on timefixed domains. The effects of domain flow, such as bifurcation delay due to domain growth and destabilization due to oscillatory domain flow, on the Eckhaus instability responsible for phase slips of spatially periodic states are analyzed with the help of both local and global stability analyses. A nonlinear phase equation describing the approach to a phaseslip event is derived. Detailed analysis of a phase slip using multiple time scale methods demonstrates different mechanisms governing the wavelength changing process at different stages.
PubDate: 20161030T22:02:28Z
DOI: 10.1016/j.physd.2016.10.003
 Authors: R. Krechetnikov; E. Knobloch
 Isolating blocks as computational tools in the circular restricted
threebody problem Authors: Rodney L. Anderson; Robert W. Easton; Martin W. Lo
Abstract: Publication date: Available online 29 October 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Rodney L. Anderson, Robert W. Easton, Martin W. Lo
Isolating blocks may be used as computational tools to search for the invariant manifolds of orbits and hyperbolic invariant sets associated with libration points while also giving additional insight into the dynamics of the flow in these regions. We use isolating blocks to investigate the dynamics of objects entering the EarthMoon system in the circular restricted threebody problem with energies close to the energy of the L 2 libration point. Specifically, the stable and unstable manifolds of Lyapunov orbits and the hyperbolic invariant set around the libration points are obtained by numerically computing the way orbits exit from an isolating block in combination with a bisection method. Invariant spheres of solutions in the spatial problem may then be located using the resulting manifolds.
PubDate: 20161030T22:02:28Z
DOI: 10.1016/j.physd.2016.10.004
 Authors: Rodney L. Anderson; Robert W. Easton; Martin W. Lo
 Microorganism billiards
 Authors: Saverio E. Spagnolie; Colin Wahl; Joseph Lukasik; JeanLuc Thiffeault
Abstract: Publication date: Available online 18 October 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Saverio E. Spagnolie, Colin Wahl, Joseph Lukasik, JeanLuc Thiffeault
Recent experiments and numerical simulations have shown that certain types of microorganisms “reflect” off of a flat surface at a critical angle of departure, independent of the angle of incidence. The nature of the reflection may be active (cell and flagellar contact with the surface) or passive (hydrodynamic) interactions. We explore the billiardlike motion of a body with this empirical reflection law inside a regular polygon and show that the dynamics can settle on a stable periodic orbit or can be chaotic, depending on the swimmer’s departure angle and the domain geometry. The dynamics are often found to be robust to the introduction of weak random fluctuations. The Lyapunov exponent of swimmer trajectories can be positive or negative, can have extremal values, and can have discontinuities depending on the degree of the polygon. A passive sorting device is proposed that traps swimmers of different departure angles into separate bins. We also study the external problem of a microorganism swimming in a patterned environment of square obstacles, where the departure angle dictates the possibility of trapping or diffusive trajectories.
PubDate: 20161023T18:28:43Z
DOI: 10.1016/j.physd.2016.09.010
 Authors: Saverio E. Spagnolie; Colin Wahl; Joseph Lukasik; JeanLuc Thiffeault
 Modulational instability in a PTsymmetric vector nonlinear
Schrödinger system Authors: J.T. Cole; K.G. Makris Z.H. Musslimani D.N. Christodoulides Rotter
Abstract: Publication date: 1 December 2016
Source:Physica D: Nonlinear Phenomena, Volume 336
Author(s): J.T. Cole, K.G. Makris, Z.H. Musslimani, D.N. Christodoulides, S. Rotter
A class of exact multicomponent constant intensity solutions to a vector nonlinear Schrödinger (NLS) system in the presence of an external P T symmetric complex potential is constructed. This type of uniform wave pattern displays a nontrivial phase whose spatial dependence is induced by the lattice structure. In this regard, light can propagate without scattering while retaining its original form despite the presence of inhomogeneous gain and loss. These constantintensity continuous waves are then used to perform a modulational instability analysis in the presence of both nonhermitian media and cubic nonlinearity. A linear stability eigenvalue problem is formulated that governs the dynamical evolution of the periodic perturbation and its spectrum is numerically determined using Fourier–Floquet–Bloch theory. In the selffocusing case, we identify an intensity threshold above which the constantintensity modes are modulationally unstable for any Floquet–Bloch momentum belonging to the first Brillouin zone. The picture in the selfdefocusing case is different. Contrary to the bulk vector case, where instability develops only when the waves are strongly coupled, here an instability occurs in the strong and weak coupling regimes. The linear stability results are supplemented with direct (nonlinear) numerical simulations.
PubDate: 20161016T12:47:09Z
 Authors: J.T. Cole; K.G. Makris Z.H. Musslimani D.N. Christodoulides Rotter
 On loops in the hyperbolic locus of the complex Hénon map and their
monodromies Authors: Zin Arai
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Zin Arai
We prove John Hubbard’s conjecture on the topological complexity of the hyperbolic horseshoe locus of the complex Hénon map. In fact, we show that there exist several nontrivial loops in the locus which generate infinitely many mutually different monodromies. Furthermore, we prove that the dynamics of the real Hénon map is completely determined by the monodromy of the complex Hénon map, providing the parameter of the map is contained in the hyperbolic horseshoe locus.
PubDate: 20160913T04:50:37Z
 Authors: Zin Arai